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/**
 * Recursive approach (brute force)
 * Time Complexity: O(4^n / √n) - Exponential due to repeated calculations
 * Space Complexity: O(n) - Recursion stack
 * 
 * @param {number} n - Number of nodes
 * @return {number} - Number of unique BSTs
 */
function numTreesRecursive(n) {
  // Base case: empty tree or single node tree
  if (n === 0 || n === 1) {
    return 1;
  }
  
  let total = 0;
  
  // Try each value as the root
  for (let i = 1; i <= n; i++) {
    // Number of unique left subtrees * Number of unique right subtrees
    const leftTrees = numTreesRecursive(i - 1);
    const rightTrees = numTreesRecursive(n - i);
    total += leftTrees * rightTrees;
  }
  
  return total;
}
 
/**
 * Memoization approach (top-down dynamic programming)
 * Time Complexity: O(n^2) - Each value of n is computed once, with n iterations for each
 * Space Complexity: O(n) - For memoization and recursion stack
 * 
 * @param {number} n - Number of nodes
 * @return {number} - Number of unique BSTs
 */
function numTreesMemo(n) {
  // Create memo array to store results
  const memo = new Array(n + 1).fill(-1);
  
  function calculate(n) {
    // If already computed, return from memo
    if (memo[n] !== -1) {
      return memo[n];
    }
    
    // Base case: empty tree or single node tree
    if (n === 0 || n === 1) {
      return 1;
    }
    
    let total = 0;
    
    // Try each value as the root
    for (let i = 1; i <= n; i++) {
      // Number of unique left subtrees * Number of unique right subtrees
      const leftTrees = calculate(i - 1);
      const rightTrees = calculate(n - i);
      total += leftTrees * rightTrees;
    }
    
    // Store result in memo
    memo[n] = total;
    return total;
  }
  
  return calculate(n);
}
 
/**
 * Tabulation approach (bottom-up dynamic programming)
 * Time Complexity: O(n^2) - Two nested loops
 * Space Complexity: O(n) - For the DP array
 * 
 * @param {number} n - Number of nodes
 * @return {number} - Number of unique BSTs
 */
function numTreesDP(n) {
  // Create DP array
  const dp = new Array(n + 1).fill(0);
  
  // Base cases
  dp[0] = 1;
  dp[1] = 1;
  
  // Fill DP array bottom-up
  for (let i = 2; i <= n; i++) {
    for (let j = 1; j <= i; j++) {
      // Number of unique BSTs with root j
      // Left subtree: j-1 nodes, Right subtree: i-j nodes
      dp[i] += dp[j - 1] * dp[i - j];
    }
  }
  
  return dp[n];
}
 
/**
 * Mathematical approach (Catalan Numbers)
 * Time Complexity: O(n) - Single loop
 * Space Complexity: O(1) - Constant extra space
 * 
 * @param {number} n - Number of nodes
 * @return {number} - Number of unique BSTs
 */
function numTrees(n) {
  // C(n) = C(n-1) * (4n - 2) / (n + 1)
  let result = 1;
  
  for (let i = 1; i <= n; i++) {
    result = result * (4 * i - 2) / (i + 1);
  }
  
  return Math.round(result); // Round to handle potential floating-point errors
}
 
// Test cases
console.log(numTrees(1));  // 1
console.log(numTrees(2));  // 2
console.log(numTrees(3));  // 5
console.log(numTrees(4));  // 14
console.log(numTrees(5));  // 42